On Compactness of Commutators of Multiplications and Fourier Multipliers

  • Nenad AntonićEmail author
  • Marin Mišur
  • Darko Mitrović


We generalise results on compactness of commutators of multiplications and Fourier multiplier operators by Cordes (J Funct Anal 18:115–131, 1975) with respect to the smoothness of multiplication function. Our prime motivation has been a particular case known as the first commutation lemma—the basic tool for defining H-measures and H-distributions. We review and improve the known results in the \(\mathrm{L}^p\) setting, with \(1<p<\infty \), illustrating these results with an application, obtaining a generalised compactness by compensation result.


Commutator compactness Fourier multiplier H-distribution 

Mathematics Subject Classification

42B15 35B40 28C15 


  1. 1.
    Aleksić, J., Pilipović, S., Vojnović, I.: H-distributions via Sobolev spaces. Mediterr. J. Math. 13, 3499–3512 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Antonić, N., Ivec, I.: On the Hörmander–Mihlin theorem for mixed-norm Lebesgue spaces. J. Math. Anal. Appl. 433, 176–199 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Antonić, N., Lazar, M.: Parabolic H-measures. J. Funct. Anal. 265, 1190–1239 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Antonić, N., Mitrović, D.: H-distributions: an extension of H-measures to an \({\rm L}^p-{\rm L}^q\) setting. Abs. Appl. Anal. 2011, Article ID 901084 (2011)Google Scholar
  5. 5.
    Cordes, H.O.: On compactness of commutators of multiplications and convolutions, and boundedness of pseudodifferential operators. J. Funct. Anal. 18, 115–131 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Edwards, R.E.: Functional Analysis: Theory and Applications. Dover, New York (1995)Google Scholar
  7. 7.
    Erceg, M., Ivec, I.: On a generalisation of H-measures. Filomat 31(16), 5027–5044 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Erceg, M., Mišur, M., Mitrović, D.: Velocity averaging and strong precompactness for degenerate parabolic equations with discontinuous flux (in preparation)Google Scholar
  9. 9.
    Gérard, P.: Microlocal defect measures. Commun. Partial Differ. Equ. 16, 1761–1794 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Grafakos, L.: Classical Fourier Analysis. Springer, Berlin (2008)zbMATHGoogle Scholar
  11. 11.
    Kohn, J.J., Nirenberg, L.: An algebra of pseudo-differential operators. Commun. Pure Appl. Math. 18, 269–305 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Krasnosel’skij, M.A.: On a theorem of M. Riesz. Dokl. Akad. Nauk SSSR 131, 246–248 (1960). [(in Russian); translated as Soviet Math. Dokl. 1 (1960) 229–231] Google Scholar
  13. 13.
    Lazar, M., Mitrović, D.: Existence of solutions for a scalar conservation law with a flux of low regularity. Electron. J. Differ. Equ. 2016(325), 1–18 (2016)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Maz’ja, V.G., Šapošnikova, T.O.: Theory of Sobolev Multipliers. Springer, Berlin (2009)Google Scholar
  15. 15.
    Mišur, M., Mitrović, D.: On a generalisation of compensated compactness in the \(\text{ L }^p\)\(\text{ L }^q\) setting. J. Funct. Anal. 268, 1904–1927 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Mišur, M., Mitrović, D.: On compactness of commutator of multiplication and pseudodifferential operator. J. Pseudo Differ. Oper. Appl. (2018). (OnlineFirst)
  17. 17.
    Murat, F.: Compacité par compensation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5, 489–507 (1978)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Panov, E.J.: Ultra-parabolic H-measures and compensated compactness. Ann. Inst. H. Poincaré Anal. Non Linéaire 28, 47–62 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Tartar, L.: H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations. Proc. R. Soc. Edinb. 115A, 193–230 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Tartar, L.: The General Theory of Homogenization: A Personalized Introduction. Springer, Berlin (2009)zbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceUniversity of ZagrebZagrebCroatia
  2. 2.Visage Technologies ABLinköpingSweden
  3. 3.Faculty of MathematicsUniversity of MontenegroPodgoricaMontenegro

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